通过动态插值研究具有粗糙数据的二维非齐次粘性流动的整体适定性
Global well-posedness for 2D inhomogeneous viscous flows with rough data via dynamic interpolation
AI总结：
针对仅具上下界密度的二维可变密度不可压缩粘性流，利用有限能量弱解衰减估计与动态插值，在临界正则性、无小性假设下证明全局存在、稳定、唯一及流映射连续。
摘要：
我们考虑具有可变密度的二维不可压缩流动的演化，该密度仅为有界且以零为下界。假设初始速度属于 L^2 的一个适当临界子空间，我们证明了初始（边）值问题的全局时间存在性和稳定性结果。我们的证明依赖于有限能量弱解的新的时间衰减估计以及“动态插值”论证。我们表明，所构造的解具有一致 C^1 流，这确保了流体中几何结构的传播，并保证方程的欧拉和拉格朗日表述等价。通过采用后一种表述，我们在类能量泛函框架中建立了给定数据解的唯一性以及流映射的连续性。与先前的工作相比，我们的结果在临界正则性设置下成立，无需任何小性假设。我们的方法仅使用基本工具，并无差别地适用于流体域为整个平面、光滑二维有界域或环面的情况。
英文摘要：
We consider the evolution of two-dimensional incompressible flows with variable density, only bounded and bounded away from zero. Assuming that the initial velocity belongs to a suitable critical subspace of L^2 , we prove a global-in-time existence and stability result for the initial (boundary) value problem. Our proof relies on new time decay estimates for finite energy weak solutions and on a 'dynamic interpolation' argument. We show that the constructed solutions have a uniformly C^1 flow, which ensures the propagation of geometrical structures in the fluid and guarantees that the Eulerian and Lagrangian formulations of the equations are equivalent. By adopting this latter formulation, we establish the uniqueness of the solutions for prescribed data, and the continuity of the flow map in an energy-like functional framework. In contrast with prior works, our results hold true in the critical regularity setting without any smallness assumption. Our approach uses only elementary tools and applies indistinctly to the cases where the fluid domain is the whole plane, a smooth two-dimensional bounded domain or the torus.
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